Show that for the curve `b y^2=(x+a)^3,` the square of the sub-tangent varies as the sub-normal.

In the curve x^(m+n)=a^(m-n)y^(2n) , prove that the m t h power of the sub-tangent varies as the n t h power of the sub-normal.

In the curve y=c e^(x/a) , the sub-tangent is constant sub-normal varies as the square of the ordinate tangent at (x_1,y_1) on the curve intersects the x-axis at a distance of (x_1-a) from the origin equation of the normal at the point where the curve cuts y-a xi s is c y+a x=c^2

Find the length of sub-tangent to the curve y=e^(x//a)

For the curve y=a1n(x^2-a^2) , show that the sum of length of tangent and sub-tangent at any point is proportional to product of coordinates of point of tangency.

A curve y=f(x) passes through point P(1,1) . The normal to the curve at P is a (y-1)+(x-1)=0 . If the slope of the tangent at any point on the curve is proportional to the ordinate of the point, then the equation of the curve is

If y=2x+3 is a tangent to the parabola y^2=24 x , then find its distance from the parallel normal.

Find the points on the curve x^(2) + y^(2) – 2x – 3 = 0 at which the tangents are parallel to the x-axis.

Show that the family of curves for which the slope of the tangent at any point (x,y) on it is (x^(2)+y^(2))/(2xy), is given by x^(2)-y^(2)=Cx

Find the equations of the tangents to the curve y=1+x^(3) for which the tangent is orthogonal with the line x + 12y = 12.